If n Is A Positive Integer, Is n – 1 Divisible By 3? GMAT Data Sufficiency

Let's take each statement one by one

S1: (n^2 + n) is not divisible by 6
n(n + 1) is not divisible by 6 (= 2 * 3)

We see that n and (n + 1) are two consecutive integers. Note that every second integer is even, thus, it is divisible by 2. Thus, Statement 1 can be rephrased as: n^2 + n is not divisible by 3. Also, note that one among the three consecutive integers is divisible by 3.

Thus, one among (n - 1)n(n + 1) is divisible by 3. Since we deduced that neither n nor (n + 1) is divisible by 3, this means that (n - 1) must be divisible by 3

Sufficient

S2: 3n = k + 3, where k is a positive multiple of 3.
3n - 3 = k
k = 3(n - 1)
(n - 1) may or may not be a multiple of 3
Insufficient

Correct option: A

(1) n^2 + n = n(n + 1)
Now, n and n + 1 are consecutive integers, so one of them is surely even.
Therefore, if n(n + 1) is divisible by 2 but not by 6, it means n(n + 1)is not divisible by 3.

n(n + 1) not divisible by 3 means that the consecutive numbers n and n + 1 are not divisible by 3.

As we have to have one multiple of 3 in every set of 3 consecutive integers, n - 1 must be divisible by 3.

Sufficient

(2) 3n = k + 3
3n - 3 = k
3(n - 1) = k
It is given that k is a multiple of 3, which can be seen from the result but n - 1 can take any value.

Insufficient

Correct option: A

Statement 1:
n^2+ n not divisible by 6
n can be 1, 4, 7 and 10.
From the above n - 1 = 0, 3, 6, 9 (All are divisible by 3)

Sufficient

Statement 2:
3n = k + 3
k can be 3, 6, 9 and 12 etc
Then n can be equal to 2, 3, 4, 5 respectively
Of the above numbers, n - 1:- 1 and 2 are not divisible by 3 but 4 - 1 is divisible by 3

Not sufficient

Correct option: A